$$f(a,x)=\sum_{\tau=-\infty}^{\infty}\frac{\exp\left(2\pi i\tau x\right)}{(\tau+a)^{p+1}}$$ How behave this function? How to interpolate it with integral? Can I somehow interpolate it with $$\int_{-\infty}^{\infty}\frac{\exp\left(2\pi i\tau x\right)}{(\tau+a)^{p+1}}d\tau$$

where $a\in(0,0.5)$ , p is a natural number, $x$ is a real number